Zeros of Polynomial Functions

1. Zeros of Polynomial Functions Advanced Math Section 3.4

2. Number of zeros • Any nth degree polynomial can have at most nreal zeros • Using complex numbers, every nth degree polynomial has precisely n zeros (real or imaginary) Advanced Math 3.4 - 3.5

3. Fundamental Theorem of Algebra • If f(x) is a polynomial of degree n, where n > 0, • then f has at least one zero in the complex number system Advanced Math 3.4 - 3.5

4. Linear Factorization Theorem • If f(x) is a polynomial of degree n, where n > 0, • then f has precisely n linear factors Advanced Math 3.4 - 3.5

5. Linear Factorization Theorem applied Advanced Math 3.4 - 3.5

6. Example • Find all zeros Advanced Math 3.4 - 3.5

7. Rational Zero Test Advanced Math 3.4 - 3.5

8. Using the rational zero test • List all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient • Use trial-and-error to determine which, if any are actual zeros of the polynomial • Can use table on graphing calculator to speed up calculations Advanced Math 3.4 - 3.5

9. Example • Use the Rational Zero Test to find the rational zeros Advanced Math 3.4 - 3.5

10. Using synthetic division • Test all factors to see if the remainder is zero • Can also use graphing calculator to estimate zeros, then only check possibilities near your estimate Advanced Math 3.4 - 3.5

11. Examples • Find all rational zeros Advanced Math 3.4 - 3.5

12. Conjugate pairs • If the polynomial has real coefficients, • then zeros occur in conjugate pairs • If a + bi is a zero, then a – bi also is a zero. Advanced Math 3.4 - 3.5

13. Example: • Find a fourth-degree polynomial function with real coefficients that has zeros -2, -2, and 4i Advanced Math 3.4 - 3.5

14. Factors of a Polynomial • Even if you don’t want to use complex numbers • Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros Advanced Math 3.4 - 3.5

15. Quadratic factors • If they can’t be factored farther without using complex numbers, they are irreducible over the reals Advanced Math 3.4 - 3.5

16. Quadratic factors • If they can’t be factored farther without using irrational numbers, they are irreducible over the rationals • These are reducible over the reals Advanced Math 3.4 - 3.5

17. Finding zeros of a polynomial function • If given a complex factor • Its conjugate must be a factor • Multiply the two conjugates – this will give you a real zero • Use long division or synthetic division to find more factors • If not given any factors • Use the rational zero test to find rational zeros • Factor or use the quadratic formula to find the rest Advanced Math 3.4 - 3.5

18. Examples • Use the given zero to find all zeros of the function Advanced Math 3.4 - 3.5

19. Examples • Find all the zeros of the function and write the polynomial as a product of linear factors Advanced Math 3.4 - 3.5

20. Descartes’s Rule of Signs • A variation in sign means that two consecutive coefficients have opposite signs • For a polynomial with real coefficients and a constant term, • The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer • The number of negative real zeros is either equal to the variations in sign of f(-x) or less than that number by an even integer. Advanced Math 3.4 - 3.5

21. Examples • Determine the possible numbers of positive and negative zeros Advanced Math 3.4 - 3.5

22. Upper Bound Rule • When using synthetic division • If what you try isn’t a factor, but • The number on the outside of the synthetic division is positive • And each number in the answer is either positive or zero • then the number on the outside is an upper bound for the real zeros Advanced Math 3.4 - 3.5

23. Lower Bound Rule • When using synthetic division • If what you try isn’t a factor, but • The number on the outside of the synthetic division is negative • The numbers in the answer are alternately positive and negative (zeros can count as either) • then the number on the outside is a lower bound for the real zeros Advanced Math 3.4 - 3.5

24. Examples • Use synthetic division to verify the upper and lower bounds of the real zeros Advanced Math 3.4 - 3.5

25. Mathematical Modeling and Variation Advanced Math Section 3.5

26. Two basic types of linear models • y-intercept is nonzero • y-intercept is zero Advanced Math 3.4 - 3.5

27. Direct Variation • Linear • k is slope • y varies directly as x • y is directly proportional to x Advanced Math 3.4 - 3.5

28. Direct Variation as an nth power • y varies directly as the nth power of x • y is directly proportional to the nth power of x Advanced Math 3.4 - 3.5

29. Inverse Variation • Hyperbola (when k is nonzero) • y varies inversely as x • y is inversely proportional to x Advanced Math 3.4 - 3.5

30. Inverse Variation as an nth power • y varies inversely as the nth power of x • y is inversely proportional to the nth power of x Advanced Math 3.4 - 3.5

31. Joint Variation • Describes two different direct variations • z varies jointly as x and y • z is jointly proportional to x and y Advanced Math 3.4 - 3.5

32. Joint Variation as an nth and mth power • z varies jointly as the nth power of x and the mth power of y • z is jointly proportional to the nth power of x and the mth power of y Advanced Math 3.4 - 3.5

33. Examples • Find a math model representing the following statements and find the constants of proportionality • A varies directly as r2. • When r = 3, A = 9p • y varies inversely as x • When x = 25, y = 3 • z varies jointly as x and y • When x = 4 and y = 8, z = 64 Advanced Math 3.4 - 3.5